Int. J. Open Problems Compt. Math., Vol. 10, No. 2, June 2017
ISSN 1998-6262; Copyright © ICSRS Publication, 2017
www.i-csrs.org
Optimization Approach Based on Immigration
Strategies for Symmetric Traveling Salesman
Problem
Chakir Tajani , Otman Abdoun
Abdelmalek Essaadi University, Morocco
e-mail: [email protected], [email protected]
JaafarAbouchabaka, Ibn TofailUniversity, Morocco
e-mail: [email protected]
Abstract
The Traveling Salesman Problem (TSP) is a combinatorial optimization problem of great importance which continues to interest several researchers in order to develop methods to achieve an optimal solution. Genetic algorithms (GAs) as meta-heuristic methods have been widely applied to this problem. Inspired by biological phenomena, we introduce two immigration operators, random immigration and structured memory immigration, forming two different algorithms. The performance of these algorithms is evaluated using benchmark datasets of symmetric TSP from TSPLIB library. The results of the proposed algorithms are compared with the standard genetic algorithm showing that the proposed algorithms improve the performance of GA in solving TSP problem effectively and specifically with the developed structured memory immigration.
Keywords:Combinatorial Problem; STSP; Genetic Algorithm; Immigration.
Optimization Approach Based on Immigration Strategies for STSP 39
1 Introduction
The Traveling Salesman Problem (TSP ) is one of the most famous problems in
combinatorial optimization which consists of solving the problem of visiting all
cities by a salesman with the condition that each city can only be visited once.
Hence, the problem is to minimize the distance of the complete tour that the
salesman may take. Several real world application of the TSP can behind, such as
Vehicle Routing [1,2], Scheduling, Image Processing and Pattern Recognition [3].
The TSP problem is classified as an NP-complete problem such [4]. This explain
the number of research and the methods developed and exploited to resolve it.
There are some methods to find the approximate solutions [5, 6], but, these
methods have exponential complexity, they take too much computing time or
require too much memory. Then, meta-heuristics methods should be exploited.
Many exact and heuristic algorithms have been proposed to solve this problem in
recent years such as branch-and-bound [7], neural network [8], [9] and artificial
bee colony [10], an ant system with a cooperating agents strategy [11], a genetic
algorithm [12],a scatter search with a particle filter strategy [13]; a particle swarm
optimization [14,15,16], ant colony optimization [17,18]. The Bat algorithm [19].
This technique does not guarantee the best solution but it is to come as close as
possible to the optimum value in a reasonable amount of time which is at most
polynomial time.
The genetic algorithm is a one of the family of evolutionary algorithms [20]. The
population of a genetic algorithm (GA) evolves by using genetic operators
inspired by the evolutionary in biology. These algorithms were modeled on the
natural evolution of species. We add to this evolution the concepts of observed
properties of genetics (Selection, Crossover, Mutation, etc), from which the name
Genetic Algorithm. They attracted the interest of many researchers, starting with
Holland [21], who developed the basic principles of genetic algorithm, and
Goldberg [22] that has used these principles to solve specific optimization
problems. Other researchers have followed this path [23,24].
In a genetic algorithm, a population of individuals (possible solution) is randomly
selected. These individuals are subject to several operations, called genetic
operators (selection, crossover, mutation, insertion,...) to produce a new
population containing in principle better individual. This population evolves more
and more until a stopping criterion is satisfied and declaring obtaining optimal
best solution. Thus; the performance of a genetic algorithm depends on the choice
of operators who will intervene in the production of the new populations
[25,26,30].
In the general structure of a genetic algorithm, we find various steps: Coding,
method of selection, crossover and mutation operator and their probabilities,
C. Tajani , O. Abdoun and J. Abouchabaka
40
insertion mechanism, and the stopping test. For each of these steps, there are
several possibilities. The choice between these various possibilities allows
creating several variants of genetic algorithms which can improve the GA. Several
works are focused on the improvement of genetic operators, which has allowed
the development of several adapted presentation, adapted crossover and mutation
operators for TSP [27] and the comparison of their performance, and even the
hybridization between two operators to benefit of their specificity and make the
GA more efficient and accurate. But, it is well known that GAs gets stuck in local
optima very often. One efficient way of avoiding this problem is maintaining the
diversification in population. Then; an immigration operator which consists in
randomly generating a finite number of individuals at regular intervals to replace a
substantial percentage of the population [28], can be applied in addition to the
usual genetic operator. In addition, knowing that the more different the
immigrants are, the more progress and knowledge is brought.
We propose an immigration technique in which the immigrants are not random,
but we adopt a technique based on structured memory immigration which consists
in benefiting individuals exclude during the previous generations. Thus, a
percentage of the most powerful individuals will immigrate after an interval of
time instead of the same number of the lowest individuals in the last generation.
The complexity of immigration is decreased by executing it only every several
generations.
In this work, we are interested in the resolution of TSP by genetic algorithm. We
will present each individual of population by the most adapted method of data
representation, the path representation method, which is the most natural
representation of a tour. The OX crossover operator and RSM mutation operator
adapted to the TSP problem are used by introducing two immigration strategies in
order to bring dynamism and then diversity to the current population to perform
the algorithm and obtain a best optimal solution in a reduced number of iteration.
This paper is organized as follows. The TSP model is presented in Section 2. The
genetic algorithm with random immigration and structured memory strategies is
prescribed. In Section 3, Computational experiments were performed through
many standard instances of TSPLIB [29] showing that the immigration strategies
perform the genetic algorithm, especially with the structured memory
immigration.
2 The Problems description
Before describing the problem, the reason behind choosing the above mentioned
TSP is that, they are very common and are derived from real world applications.
Hence, the TSP provide a very good platform for testing the impact of an elite
pool on the performance and generality (consistency and efficiency) of our
proposed genetic algorithm.
Optimization Approach Based on Immigration Strategies for STSP 41
The Traveling Salesman Problem (TSP) consist to finding the shortest closed
route among n cities, having as input the complete distance matrix among all
cities.
Let dij be a non-negative integer that stands for the cost (distance) to travel from
city i directly to city j.
If dij = dji the problem is called symmetric traveling salesman problem (STSP).
The STSP subject of this paper can formally be stated as follows:
Given a set of cities and the cost of travel (or distance) between each possible
pairs, the TSP, is to find the best possible way of visiting all the cities and
returning to the starting point that minimize the travel cost (or travel distance).
The objective is to find a tour of the minimum total length, where the length is the
sum of the costs of each arc in the tour.
The search space for the STSP is a set of permutations of n cities and the optimal
solution is a permutation which yields the minimum cost of the tour.
In other words, an STSP of size nis defined by:
We consider a set of points 𝑣 = {𝑣1, 𝑣2, … . , 𝑣𝑛} which 𝑣𝑖 is a city.
The mathematical formulation of the ATSP is given by:
𝑀𝑖𝑛 𝑓 𝑇 , 𝑇 = 𝑇 1 , 𝑇 2 , … . . , 𝑇 𝑛 , 𝑇 1 (1),
Where;f is the evaluation function calculates the adaptation of each solution of the
problem given by the following formula:
𝑓 = 𝑑 𝑇(𝑖), 𝑇(𝑖 + 1) + 𝑑 𝑇(𝑛), 𝑇(1) 𝑖=𝑛−1𝑖=1 (2),
With d define a symmetric distance matrix and d(i, j) the distance between the city
𝑣𝑖 and 𝑣𝑗 .
𝑇[𝑖]is a permutation on the set {1, 2, . . . ,n}.T = (T[1], T[2], . . . . . . , T[n], T[1])
is the scheduling form of a solution of the STSP.
In this study, we adopt the concept of calculating the distances between cities
from [20], where the distance between city i and city (i + 1) is calculated as the
Euclidean distance using the following equation:
𝑑 𝑇 𝑖 , 𝑇 𝑖 + = (𝑥𝑖 − 𝑥𝑖+1)2 + (𝑦𝑖 − 𝑦𝑖+1)2 (3),
C. Tajani , O. Abdoun and J. Abouchabaka
42
3 Proposed Immigration Approach
In this work, we consider the resolution of the TSP by Genetic Algorithms where
we will present each individual by the most adapted and natural method of data
representation, the path representation method (fig. 1), which is the most natural
representation of a tour (a tour is encoded by an array of integers representing the
successor and predecessor of each city).
Fig. 1. Coding of a tour (3 5 2 9 7 6 8 4)
In this work, in the standard genetic algorithm, a population of individuals
represented by "Path representation" is chosen at random and evaluated to select
individuals who will undergo a set of genetic operators, namely, “OX Crossover”
[25] and “RSM Mutation” [26]. Then, evaluation is maid for the new individuals
(offspring’s) to proceed to the insertion to give birth to a new population by
serving the best individual of the old population by elitism, to give birth to a new
population which will undergo the same procedure.
In the proposed approach, we introduce two strategies in the GA; the first is called
"Random Immigration" after every generation and the second is called "Structured
Memory Immigration" after a finite number of generations.
3.1 Crossover operator - OX -
The crossover in carried out according to the crossover probability Px. In this
study, we chose as crossover operator the Ordered Crossover method (OX) [25].
The Ordered Crossover method is presented by Goldberg [22], is used when the
problem is of order based, for example in U-shaped assembly line balancing etc.
Given two parent chromosomes, two random crossover points are selected
partitioning them into a left, middle and right portion. The ordered two points
crossover behaves in the following way: child1 inherits its left and right section
from parent1, and its middle section is determined.
We have chosen the operator OX as a crossover operator in this study because it is
considered one of the best genetic operators used in the resolution of the traveling
salesman problem [25].
Optimization Approach Based on Immigration Strategies for STSP 43
Fig.2. Algorithm of crossover operator OX
3.2 Mutation operator - RSM -
In the reverse sequence mutation operator (RSM) [26], we take a sequence S
limited by two positions i and j randomly chosen, such that 𝑖 < 𝑗. The gene order
in this sequence will be reversed by the same way as what has been covered in the
previous operation. The algorithm (Fig. 3) shows the implementation of this
mutation operator with example in (Fig 4.).
Fig.3. Algorithm of RSM operator
Parent 1 2 3 4 5 6 Child 1 5 4 3 2 6
Fig.4. Mutation operator RSM
Input: Parents x1=[x1,1,x1,2,……,x1,n] and x2=[x2,1,x2,2,……,x2,n]
Output:Children y1=[y1,1,y1,2,……,y1,n] and y2=[y2,1,y2,2,……,y2,n]
-----------------------------------------------------------
Choose two crossover points a and b such that 1≤a≤b≤n;
Repeat Permute (xa, xb);
a = a + 1;
b = b − 1;
until a<b
Input: Parents x1=[x1,1,x1,2,……,x1,n] and x2=[x2,1,x2,2,……,x2,n]
Output:Children y1=[y1,1,y1,2,……,y1,n] and y2=[y2,1,y2,2,……,y2,n]
---------------------------------------------------------
Initialize
Initialize y1 and y2 being a empty genotypes;
Choose two crossover points a and b such that 1≤a≤b≤n;
j1 = j2 = k = b+1;
i = 1;
Repeat
if x1,i {x2,a, . . . ,x2,b} then
{
y1,j1 = x1,k ;
j1++;
}
if x2,i {x1,a, . . . ,x1,b} then
{
y2,j1 = x2,k ;
j2++;
}
k=k+1;
Untili ≤ n
y1 = [y1,1 ……y1,a−1 x2,a ……x2,b y1,a ……y1,n−a];
y2 = [y2,1 ……y2,a−1 x1,a ……x1,b y2,a ……y2,n−a];
C. Tajani , O. Abdoun and J. Abouchabaka
44
3.3 Immigration strategies
In addition to the standard operators of a genetic algorithm, namely the crossover
and the mutation, we introduce a new operator called "Immigration Operator"
which will introduce a diversity of the population and then more dynamism and
exploration of different probable solutions of the problem, In order to obtain a
better optimal solution compared to that obtained by the standard genetic
algorithm and for the genetic algorithm based on the immigration operator.
Two strategies are proposed, the first one is called “Random Immigration” where
the randomly created individuals are inserted into the population every generation
by replacing the worst individuals or some individuals randomly selected
individual (Fig.5.).
Fig.5. Random immigration genetic algorithm - RIGA-
However, in order to benefit of the previous generations and of some individuals
that not be able to be introduced in the population. After a defined interval of time
(some generation), we give chance of the best individual to immigrate to the new
population. This new operator is called "Structured Memory Immigration
Operator" which is the second strategy (Fig. 6.).
This way, the introduced immigrants are more adapted to the current environment
than the random immigrants. Then, the new operator introduces a diversity of the
population and more dynamism and exploration of different probable solutions of
the problem, in order to obtain a better optimal solution compared to that obtained
by the standard genetic algorithm and for the genetic algorithm based on the
immigration operator.
begin
Initialize population P randomly with constraints validation
evaluate population P
for (iitr=1; iitr<=iter; iitr++)
Sel:=Select For Reproduction(P) // N individuals
CX := Crossover(Sel, Px) // Px is the crossover probability
Mut := Mutate(CX, Pm) // Pm is the mutation probability
Evaluate new individuals Mut // EvaluteMut and sort in descendant
P’ = Elitisme(Mut(1; N/2)) // Perform elitism
If mod( iitr, ItInser) == 0 then // Execution of RIGA
Generate n random immigrants
evaluate these random immigrants
replace the worst individuals in P
endIf
endfor
end
Optimization Approach Based on Immigration Strategies for STSP 45
Fig.6. Structured memory immigration genetic algorithm -SMIGA-
4 Experimental results
4.1 Implementation and instances
The algorithms are coded in C++ and run on a PC HP Intel Corei31.7 GHz and
4GB of RAM.
In this article, the proposed genetic algorithm using the two strategies for the
resolution of TSP is tested on two categories of STSP-instances: One category
(C1) with a limited number of breakpoints which does not exceed a maximum of
42 jumps and the other (C2) with a wider length exceeding 100 breakpoints.
Results are presented separately, starting with those obtained by small TSP
instances: BURMA14, RG17, ULYSSES22, FRI26, MATBAYS29 and
DANTZIG42. Then, the performance of the proposed approachis tested with large
instances: ATT48, ST70, PR76, KROA100 and EIL101. All instances used in
numerical tests are standard instances of the TSP library [29]. The effectiveness of
the Immigration Approach (AIG) “Structured Memory Immigration Algorithm”,
proposed in this paper, is demonstrated through comparisons with a Variant
Genetic Algorithm (VGA) and Standard Immigration (SIG) called here “Random
immigration genetic algorithm”.
The genetic parameters of the AIG, VGA and SIG used to obtain the results
presented are: Npop = 10, Select: Roulette [25], Niter = 100, Px = 0.6, Pm ∈
[0.001, 0.2] and Insert: Elitism. They were chosen after preliminary tests, in order
to achieve a good compromise between calculation time and the quality of the
solutions.
Indeed, the time to build a solution is not negligible; the total number of iterations
must remain quite small. Diversification explores the space of solutions in order
to search for a subset of individuals to obtain good results.
begin
Initialize population P randomly with constraints validation;
evaluate population P;
for(iitr=1; iitr<=iter; iitr++)
Sel:=Select For Reproduction (P) // N individuals
CX := Crossover (Sel, Px) // Px is the crossover probability
Mut := Mutate (CX, Pm) //Pm is the mutation probability
Evaluate new individuals Mut // EvaluteMut and sort in descendant
P’ = Elitisme(Mut(1; N/2)) // Perform elitism
ImPop = (Mut(N/2; N))
ifiitr%ItInsert == 0 then // Execution of SMIGA
evaluate immigrants subpopulation ImPop
replace the n worst individuals in P
endif
endfor
end
C. Tajani , O. Abdoun and J. Abouchabaka
46
The results are given in table 1 and table 2. The first column indicates the TSP
instances used. The other characteristics appearing in the tables present the results
were given on average per subgroup of instances of equivalent size, for the
proposed AIG method comparing it with other optimization methods: VGA and
SIG. The two columns Init and Bsol expose respectively the initial solution and
the optimal solution obtained after a well defined number of iterations.
4.2 Results for the first group of instances - C1-
Table 1. shows the numerical values found after execution of the new approach
proposed in this paper and the similar values generated by standard immigration
(SIG) and the VGA genetic variant on small (C1) TSP instances. The results show
that the meta-heuristic immigration AIG improves those of the heuristic VGA and
also SIG. The gain can even be quite significant, respectively 23.52% and also
10.34% on a small instance like Burma14.
Thus, the effectiveness of the AIG method is sufficiently influential to induce a
difference of convergence towards the optimal solution (24.36%). More precisely
for the instance RG17, by way of indication, the solution of the VGA method is of
a total cost equal to 3407 against 2577 in the solution of the AIG approach. On the
MATBAYS29 instance, the AIG finds 15528 as the optimum solution compared to
16797 provided by SIG.
With the application of the three methods VGA, SIG and AIG to the resolution of
the first category (C1) of the small size TSP instances: BURMA14, RG17,
ULYSSES22, FRI26, MATBAYS29 and DANTZIG42 is represented respectively in
fig.6(a), fig.6(b), fig.6(c), fig.6(d), fig.6(e) and fig.6(f).
Figure 6. illustrates the evolution of the cost function with respect to the number
of iterations. The curves of the graph synthesize the average performance of the
proposed method to solve the instances of type C1. Besides the fact that this
figure highlights the need for AIG genetic immigration to obtain good quality
solutions, according to the requirements retained, the AIG method can provide the
optimal solution in a very reduced number of iterations.
Table 1. The obtained solution for small size TSP instances after 100 iterations
VGA SIG AIG
Init Bsol Init Bsol Init Bsol
BURMA14 47 34 50 29 48 26
RG17 3941 3407 4629 2822 3975 2577
ULYSSES22 150 104 144 95 144 86
FRI26 2390 1871 2308 1776 2399 1695
MATBAYS29 22688 19100 22569 16797 24412 15528
DANTZIG42 2722 2135 2605 2028 2729 1926
Optimization Approach Based on Immigration Strategies for STSP 47
Fig.6. Convergence to the global optimum for small TSP instances:(a) TSP-
BURMA14;(b) TSP-RG17; (c) TSP-ULYSSES22; (d) TSP-FRI26; (e) TSP-
MATBAYS29; (f)TSP-DANTZIG42.
(a)
(b)
(c)
(d)
(e)
(f)
C. Tajani , O. Abdoun and J. Abouchabaka
48
4.3 Results for the second group of instances - C2 -
To test the function of the proposed approach in the resolution of ATSP instances
of different sizes, we have varied the deployment of the proposed approach with
large (C2) instances, comparing its generated numerical result with the other VGA
approaches and SIG.
Table 2. shows that AIG exceeds SIG and VGA performance in terms of cost.
Once again, if on average the number of breakpoints in a TSP business traveler
problem increases, can reach more than 100 jumps, a more careful observation
shows that an increase in the size of the TSP instance does not necessarily
influence the convergence of the proposed AIG approach to optimize the total cost
of the solution.
With regard to calculation times, no method requires more than one second with
less than 40 jumps. On the other hand, for the biggest instances, if VGA obtains
results in less than 10 s, the immigration approach can sometimes require several
seconds more. However, as in previous instances, the gain on cost is interesting
and the calculation times are reasonable for strategic decisions.
The fig.7emphasize the effectiveness of the proposed AIG approach in solving all
variants of the ATSP problem with even large instances, fig.7(e) and fig.7(f), in a
reasonableiteration number.
The numerical results emphasize the importance of the proposed approach,
consisting of a good selection of genetic operators enhanced by the integration of
the improved immigration technique, which results in a good compromise
between the total cost and the number of iteration in order to find the optimal
solution.
The goal is not necessarily to try to reduce the execution CPUtime, but to find a
good compromise between the number of iterations executed by the genetic
approach and the values of the fitness function, so as to find a combination
minimizing the total cost of the optimal solution.
The best quality of the solutions obtained by the AIG is however to the detriment
of the average computing times which go from 6 seconds for VGA to 17 seconds.
It should be noted, however, that the decisions taken are of a strategic type, in the
long term, and therefore that the gains made deserve a few extra seconds.
Table 2. Values of optimum TSP instances of large size after 100 iterations
VGA SIG AIG
Init Bsol Init Bsol Init Bsol
ATT48 151923 109174 139863 94492 140537 91776
ST70 3477 2948 3411 2496 3498 2378
PR76 546786 471677 538220 393494 539447 379843
KROA100 163133 130308 158079 119650 158725 113962
EIL101 3158 2595 3187 2471 3269 2383
Optimization Approach Based on Immigration Strategies for STSP 49
Fig.7. Convergence to optimum for large TSP instances:(a)TSP-ATT48; (b) TSP-
ST70; (c) TSP-PR76; (d) TSP-KROA100; (e)TSP-EIL101.
(a)
(b)
(c)
(d)
(e)
C. Tajani , O. Abdoun and J. Abouchabaka
50
5 Conclusion
In this work, in order to improve the performance of genetic algorithm to solve
the symmetric traveling salesman problem as a typical problem of great
importance in the real world. In addition to the standard operations of a GA, two
immigration strategies are introduced "standard immigration" and "structured
memory immigration". The results obtained with the standard instances of the
TSPLIB show the operators effectiveness in bringing diversity to the genetic
population and dynamism. Thus, thereafter, the performance of the solution even
after a finite number of iterations and especially with the proposed operator
"structured memory migration" which allows us to benefit, after a number of
iterations, better old individuals who could not be inserted into previous
populations.
6 Open Problem
Inspired by natural phenomena, we have introduced the immigration strategy that
has improved the performance of the genetic algorithm. In future work, we want
to introduce this strategy to other extensions of the TSP problem and especially to
the DSTP. On the other hand, the adaptation of genetic algorithms to other
transport problems can be envisaged with new representations of some transport
problems or the coding of solutions to a great influence on the performance of
AGs.
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